Asymptotic behavior for dissipative Korteweg-de Vrie equations

TL;DR

This paper investigates the long-term behavior of solutions to dissipative Korteweg-de Vries equations, identifying decay rates and asymptotic profiles in Sobolev spaces for different dissipation levels.

Contribution

It provides a detailed analysis of the asymptotic decay and convergence rates of solutions to the dissipative KdV equations with fractional dissipation.

Findings

Solutions decay like t^{-r(α)} in Sobolev norms

Identifies specific decay rates depending on α

Establishes asymptotic profiles for large time behavior

Abstract

We study the large time behavior of solutions to the dissipative Korteweg-de Vrie equations $u_t+u_{xxx}+|D|^{\alpha}u+uu_x=0$ with $0<\alpha<2$. We find $v$ such that $u-v$ decays like $t^{-r(\alpha)}$ as $t\to\infty$ in various Sobolev norm.